A246185 Triangle read by rows: T(n,k) is the number of weighted lattice paths B(n) for which the area between the path and the lines y=0 and y=1 is equal to k. B(n) is the set of lattice paths of weight n that start in (0,0), end on the horizontal axis and never go below this axis, whose steps are of the following four kinds: an (1,0)-step of weight 1; an (1,0)-step of weight 2; a (1,1)-step of weight 2; a (1,-1)-step of weight 1. The weight of a path is the sum of the weights of its steps.

1, 1, 2, 3, 1, 5, 2, 1, 8, 5, 3, 1, 13, 10, 8, 5, 1, 21, 20, 18, 14, 8, 1, 34, 38, 39, 35, 26, 12, 1, 55, 71, 80, 80, 70, 49, 17, 1, 89, 130, 160, 174, 169, 142, 90, 23, 1, 144, 235, 312, 365, 385, 363, 290, 158, 30, 1, 233, 420, 598, 745, 840, 861, 785, 588, 264, 38, 1

Offset: 0

Author

Emeric Deutsch, Aug 28 2014

Keywords

• nonn
• tabf

Comments

Number of entries in row n is n-1 (n>=2).

Sum of entries in row n is A004148(n+1) (the 2ndary structure numbers).

T(n,0) = A000045(n+1) (the Fibonacci numbers).

T(n,1) = A001629(n-1). - Robert Israel, Aug 28 2014

Examples

			Row 3 is 3, 1. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the four paths of weight 3 are ud, hH, Hh, and hhh; the areas to be considered are 1, 0, 0, and 0, respectively.
Triangle starts:
1;
1;
2;
3, 1;
5, 2,  1;
8, 5,  3, 1;
13,10, 8, 5, 1;
...
		

Links

• Alois P. Heinz, Rows n = 0..150, flattened
• M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.

Crossrefs

Cf. A004148, A000045, A246177.

Programs

Maple

eq := (t*z^3+z^2-t*z^2+z-t*z-1+t)*g^2+(t*z^2+t*z+1-2*t)*g+t = 0: g := RootOf(eq, g,1): gser := simplify(series(g, z = 0, 20)): for n from 0 to 17 do P[n] := sort(coeff(gser, z, n)) end do: 1; 1; for n from 2 to 17 do seq(coeff(P[n], t, j), j = 0 .. n-2) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, y) option remember; `if`(y<0 or y>n, 0, `if`(n=0, 1,
      expand(b(n-1, y)*x^min(1, y)+`if`(n>1, b(n-2, y)*x^min(1, y)+
      b(n-2, y+1)*x^min(y+1/2, 1), 0)+b(n-1, y-1)*x^min(y-1/2, 1))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..20);  # Alois P. Heinz, Aug 28 2014

Mathematica

b[n_, y_] := b[n, y] = If[y<0 || y>n, 0, If[n==0, 1, Expand[b[n-1, y] * x^Min[1, y] + If[n>1, b[n-2, y]*x^Min[1, y] + b[n-2, y+1]*x^Min[y+1/2, 1], 0] + b[n-1, y-1]*x^Min[y-1/2, 1]]]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 08 2017, after Alois P. Heinz *)

Formula

G.f. g = g(t,z) satisfies (t*z^3 + z^2 - t*z^2 + z - t*z - 1 + t)*g^2 +(1 - 2*t + t*z + t*z^2)*g + t = 0.

The above equation for g = G(t,1,z) follows from the trivariate g.f. G(t,w,z), where z marks weight, t marks the considered area, and w marks the length of the path (= number of steps) and which satisfies G(t,w,z) = 1 + w*z*G(t,w,z) + w*z^2*G(t,w,z) + t*w^2*z^3*G(t,w,z)*G(1,tw,z).